Subgroup ($H$) information
| Description: | $C_3^6.A_4^2:C_2^2$ |
| Order: | \(419904\)\(\medspace = 2^{6} \cdot 3^{8} \) |
| Index: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Generators: |
$\langle(2,9,3)(4,6,13)(16,17,18), (1,5,14)(11,12,15)(16,17,18), (5,14)(17,18), (16,17,18) \!\cdots\! \rangle$
|
| Derived length: | $4$ |
The subgroup is characteristic (hence normal), nonabelian, and solvable. Whether it is a direct factor, a semidirect factor, or monomial has not been computed.
Ambient group ($G$) information
| Description: | $C_3^7.A_4^2:D_4$ |
| Order: | \(2519424\)\(\medspace = 2^{7} \cdot 3^{9} \) |
| Exponent: | \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
| Derived length: | $4$ |
The ambient group is nonabelian and solvable. Whether it is monomial has not been computed.
Quotient group ($Q$) structure
| Description: | $S_3$ |
| Order: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Exponent: | \(6\)\(\medspace = 2 \cdot 3 \) |
| Automorphism Group: | $S_3$, of order \(6\)\(\medspace = 2 \cdot 3 \) |
| Outer Automorphisms: | $C_1$, of order $1$ |
| Derived length: | $2$ |
The quotient is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), hyperelementary for $p = 2$, and rational.
Automorphism information
Since the subgroup $H$ is characteristic, the automorphism group $\operatorname{Aut}(G)$ of the ambient group acts on $H$, yielding a homomorphism $\operatorname{res} : \operatorname{Aut}(G) \to \operatorname{Aut}(H)$. The image of $\operatorname{res}$ on the inner automorphism group $\operatorname{Inn}(G)$ is the Weyl group $W = G / Z_G(H)$.
| $\operatorname{Aut}(G)$ | $C_3^6.C_2^6:S_3^3$, of order \(10077696\)\(\medspace = 2^{9} \cdot 3^{9} \) |
| $\operatorname{Aut}(H)$ | $C_3^6.C_2\wr S_3^2$, of order \(1679616\)\(\medspace = 2^{8} \cdot 3^{8} \) |
| $W$ | $C_3^6.A_4^2:D_4$, of order \(839808\)\(\medspace = 2^{7} \cdot 3^{8} \) |
Related subgroups
| Centralizer: | not computed |
| Normalizer: | $C_3^7.A_4^2:D_4$ |
Other information
| Number of conjugacy classes in this autjugacy class | $1$ |
| Möbius function | not computed |
| Projective image | $C_3^7.A_4^2:D_4$ |